126 research outputs found
A Herglotz wavefunction method for solving the inverse Cauchy problem connected with the Helmholtz equation
AbstractThis paper is concerned with the Cauchy problem connected with the Helmholtz equation. On the basis of the denseness of Herglotz wavefunctions, we propose a numerical method for obtaining an approximate solution to the problem. We analyze the convergence and stability with a suitable choice of regularization method. Numerical experiments are also presented to show the effectiveness of our method
Recovering source location, polarization, and shape of obstacle from elastic scattering data
We consider an inverse elastic scattering problem of simultaneously
reconstructing a rigid obstacle and the excitation sources using near-field
measurements. A two-phase numerical method is proposed to achieve the
co-inversion of multiple targets. In the first phase, we develop several
indicator functionals to determine the source locations and the polarizations
from the total field data, and then we manage to obtain the approximate
scattered field. In this phase, only the inner products of the total field with
the fundamental solutions are involved in the computation, and thus it is
direct and computationally efficient. In the second phase, we propose an
iteration method of Newton's type to reconstruct the shape of the obstacle from
the approximate scattered field. Using the layer potential representations on
an auxiliary curve inside the obstacle, the scattered field together with its
derivative on each iteration surface can be easily derived. Theoretically, we
establish the uniqueness of the co-inversion problem and analyze the indicating
behavior of the sampling-type scheme. An explicit derivative is provided for
the Newton-type method. Numerical results are presented to corroborate the
effectiveness and efficiency of the proposed method.Comment: 29 pages, 11 figure
Co-inversion of a scattering cavity and its internal sources: uniqueness, decoupling and imaging
This paper concerns the simultaneous reconstruction of a sound-soft cavity
and its excitation sources from the total-field data. Using the single-layer
potential representations on two measurement curves, this co-inversion problem
can be decoupled into two inverse problems: an inverse cavity scattering
problem and an inverse source problem. This novel decoupling technique is fast
and easy to implement since it is based on a linear system of integral
equations. Then the uncoupled subproblems are respectively solved by the
modified optimization and sampling method. We also establish the uniqueness of
this co-inversion problem and analyze the stability of our method. Several
numerical examples are presented to demonstrate the feasibility and
effectiveness of the proposed method.Comment: 21 pages, 7 figure
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